Volume 09 - Metallography and Microstructures Part 3 ppsx

Fig. 27 Fig. 28 Fig. 29 Oriented dislocation arrays in thin foils. Fig. 27: copper. (Ref 15). Fig. 28: iron. (Ref 15). Fig. 29: Armco iron. (Ref 18). Etchants and magnifications not reported For a partially oriented system of lines in the alloy: (L V ) ran = 2(P A ) P (Eq 9a) (L V ) or = (P A ) ⊥ - (P A ) P (Eq 9b) (L V ) tot = (P A ) ⊥ + (P A ) P (Eq 9c) where (P A ) ⊥ and (P A ) P refer to measurements of point density on planes perpendicular and parallel to the orientation direction, respectively. If the system of lines is completely oriented, (P A ) P is zero, and (L V ) or = (P A ) ⊥ . Although some microstructural features are not truly linear, they can be considered so for practical purposes if they have sufficient linearity. Of course, if the cross-sectional thickness is too great, the (P A ) P measurements will be difficult to make. Another major type of oriented structure consists of surfaces in the alloy. Examples of oriented planar features in the microstructure are pearlites in steel (Fig. 30), lamellae in unidirectionally solidified eutectics (Fig. 31), and lamellar precipitates observed by thin-foil electron transmission microscopy (Fig. 32). Fig. 30 Replica electron micrograph showing lamellar pearlite in a 1090 hot-rolled steel bar. Picral. 2000× Fig. 31 Lamellae in a unidirectionally solidified aluminum-copper eutectic alloy. Etchant and magnification not reported. (Ref 1) Fig. 32 Thin-foil transmission electron micrograph showing lamellar precipitate in Fe-30Ni- 6Ti alloy. Magnification not reported. (R.C. Glenn) These oriented surfaces are subclassified as planar orientation, because the planar surfaces are essentially parallel to an orientation plane (or planes). The three equations applicable to a partially oriented system of surfaces with planar orientation are: (S V ) ran = 2(P L ) P (Eq 10a) (S V ) or = (P L ) ⊥ - (P L ) P (Eq 10b) (S V ) tot = (P L ) ⊥ + (P L ) P (Eq 10c) where (P L ) ⊥ and (P L ) P are measurements made perpendicular and parallel to the orientation plane, respectively. If the system of surfaces is completely oriented, as in portions of Fig. 32, (P L ) P is zero, and (S V ) or = (P L ) ⊥ . A sequence of extruded beryllium specimens with different initial powder sizes exemplifies the analysis of a system of partially oriented surfaces (Ref 1). The essential data are as follows: Specimen Initial powder size, mm (P L ) ⊥ , mm -1 (P L ) P mm -1 1 0.004 164.8 115.0 2 0.100 104.2 69.5 3 0.250 69.0 56.8 Assuming planar orientation, substitution in Eq 10b and 10c shows that for specimens 1, 2, and 3, respectively: (S V ) or = (P L ) ⊥ - (P L ) P = 164.8 - 115.0 = 49.8 mm -1 = 104.2 - 69.5 = 34.7 mm -1 = 69.0 - 56.8 = 12.2 mm -1 and (S V ) tot = (P L ) ⊥ + (P L ) P = 164.8 + 115.0 = 279.8 mm -1 = 104.2 + 69.5 = 173.7 mm -1 = 69.0 + 56.8 = 125.8 mm -1 The fractional, or percentage, amount of planar orientation, represented by Ω pl is (S V ) or /(S V ) tot , or: for specimens 1, 2, and 3, respectively. The results suggest that some mechanical properties may fall out of sequence even though the mean grain intercept length (equal to the reciprocal of P L ) varies directly with the initial powder size. Where the grains (or particles, inclusions, or precipitates) are markedly elongated, a shape index may prove useful. One of the simplest indices to express elongation is the ratio of mean length to mean width: () () IL LI DP Q DP ⊥ ⊥ == (Eq 11) Using the data given above for extruded beryllium specimens, Eq 11 becomes: for specimens 1, 2, and 3, respectively. For equiaxed grains, of course, Q-ratios closer to unity would be expected. Lamellar structures perhaps most typically exemplify oriented surfaces. A measure of the fineness of lamellae (as in pearlite, for example) is the so-called interlamellar or true spacing, S o , defined as the perpendicular distance across a single pair of contiguous lamellae. Because the true spacing is difficult to determine directly, the mean random spacing, σ, defined as: 1 L N σ = (Eq 12) where N L is the number of alternate lamellae intersected per unit length of random test lines, is measured instead. The true spacing can then be found according to Eq 13, which has been confirmed experimentally: 2 o S σ = (Eq 13) Figure 33 illustrates three types of spacings and three types of distances. Spacings are essentially center-to-center lengths; distances, edge-to-edge lengths. The interlamellar distances are related to the spacings through the linear intercept ratio (L L ) as in: λ= (L L ) σ (Eq 14) or by the mean intercept length (L 3 ) as in: L 3 = σ- λ (Eq 15) The subscript 3 refers to the three-dimensionality of the parameter. Finally, it is noted that: 4 v S σ = (Eq 16) where S V is the lamellar interface area per unit volume. Fig. 33 Schematic presentation of three types of spacings and three types of distances in a lamellar st ructure. (Ref 19) Grain Size Grain sizes, or diameters, have been determined by several methods. Because the grains normally found in alloys have irregular shapes, the definition of a diameter is usually arbitrary. Fortunately, a general, quantitative length parameter provides a unique, assumption-free value for any granular, space- filling structure, regardless of grain shape, size, or position. This length parameter is the mean intercept length L 3 obtained from simple L 2 intercept measurements on the plane of polish. For many random planes, of course, the averaged L 2 values become the true, three-dimensional L 3 parameter. For space-filling grains, the mean intercept length is defined as: 3 1 T L L L NPM == (Eq 17) where N L has been described above. In essence, L 3 equals the total test-line length, L T , divided by the magnification, M, and the number of grain-boundary intersections, P (P equals N for space-filling grains). When L 3 is expressed in millimeters, it gives the same value as the intercept procedure described in ASTM specification E 112 (Ref 4). This specification also is the basis for the ASTM grain-size number N, defined as: log 1.0000 log2 n N =+ (Eq 18) where n is the number of grains per square inch at 100× (n is equal to N A in the notation of this article). Normally, to obtain the ASTM grain-size number, at least 50 grains in each of three areas must be counted, the number per square inch must be determined, and this value must be converted to its equivalent at 100×. Then, substitution in Eq 18 or recourse to tables gives ASTM N. A particularly quick and useful method for determining an equivalent ASTM N uses the P L count (Ref 20). Provided are two circular test figures of known lengths, as depicted in Fig. 34 (not shown to size). The test circles can be reproduced on plastic sheet (for analyzing photomicrographs) or placed on the ground glass screen of a metallograph. The best method is to use the test circle as a reticle in the focusing eyepiece of a bench microscope. Fig. 34 Hilliard's circular test figures for measurement of grain size. The size of the circles indicated here is suitable for the ground-glass screen of a metallograph. The operator selects one of the circles and a magnification for the specimen that will result in more than six intersections per application of the circle, on the average. For equiaxed grains that do not vary much in size, the circle is applied to the microstructure until about 35 intersections are obtained, ensuring that a standard deviation of 0.3 units in G, the equivalent grain-size number, is obtained. To calculate G, the equation is: G = - 10.00 - 6.64 log L 3 (cm) (Eq 19) with: 3 T L L PM = (Eq 20) where P is the total number of grain-boundary intersections made by a test circle laid down several times to give a total length, L T (in centimeters), on a field viewed at any magnification, M. To demonstrate the operation of Eq 19, suppose that a 10-cm (4-in.) circle is applied four times to a microstructure at 250×, totaling 36 intersections. G then equals -10 - 6.64 log [40/(36 × 250)] or 5.6. Thus, the equivalent grain-size number is obtained directly and efficiently, because no more intersections are counted than required to ensure the desired accuracy. A nomograph for the graphical solution of Eq 19 is reproduced in Fig. 35. Fig. 35 Nomograph for obtaining ASTM grain-size numbers. (Ref 20) Particle Relationships Many of the relationships pertaining to particulate structures apply with equal validity to second-phase regions, voids, and boundary precipitates. One important general relationship involves the mean free distance, λ, which is the mean edge-to- edge distance, along random straight lines, between all possible pairs of particles (Ref 1). For α-phase particles, the mean free distance is: 1() Va L V N λ − = (Eq 21) where (V V )α is the volume fraction of the αparticles and N L is the number of particle interceptions per unit length of test line. Equation 21 is valid regardless of size, shape, or distribution of the particles and represents a truly three-dimensional interparticle distance. This parameter is important for studies of strength and other mechanical properties and has been used in several different ways as indicated in Fig. 36 and 37. Fig. 36 Yield strength of steels as a function of the mean free distance between cementite particles. (Ref 21) Fig. 37 Strain rate of copper-aluminum dispersion alloys as a function of mean free distance between particles. (Ref 22) There are other types of interparticle distance and spacing parameters, such as the mean particle spacing, σ, which is essentially the mean particle center-to-center length. The defining equation for mean particle spacing is: 1 L N σ = (Eq 22a) where N L is the number of particle interceptions per unit length of random test lines. The parameter σ is characterized by how easily it can be measured, because only a simple particle interception count is needed. It is also related to λ through the mean intercept length, L 3 , by the equation: λ= σ- (L 3 )α (Eq 23) where (L 3 )α for particles of αphase is defined by (L 3 )α = (L L )α/N L . This is a general and assumption-free expression, valid for particles of an size or configuration. The mean particle intercept length, (L 3 )α, is a companion term to λ, in that λis the mean matrix intercept distance and (L 3 )α is the mean particle intercept distance. They are related through the expression for a two-phase or particulate structure of αphase by: 3 1() () () Va a Va V L V λ  − =   (Eq 24) where λis the mean free distance between particles that have a volume fraction (V V )α and mean intercept length (L 3 )α. Equation 24 has been used to verify the value of volume fraction in a two-phase alloy, such as in Fig. 38. Size and configuration of the dark second phase can be varied readily by heat treatment, but the volume fraction remains relatively constant. Therefore, the (constant) volume fraction obtained from the slope of the curve for λversus (L 3 )α (73.2 vol%) corresponds well with the volume fractions determined by point counting (73.5 vol%) and from chemical analysis (71.4 vol%). Fig. 38 Interpenetrating two-phase beryllium-aluminum alloy. Etchant and magnification not reported Note that the mean intercept lengths for space-filling grains and for particles are related through the general expression: 3 L L L L N = (Eq 25) In single-phase alloys, L L (or V V ) = 1, and Eq 17 is obtained. For two-phase or particulate alloys, L L (or V V ) has a value less than 1, and Eq 25 is used. Also, 2N L = P L applies to particulate systems, whereas N L = P L applies to the single-phase alloys. An example of the application of the mean intercept lengths is seen in the well-known relationship: 4 3 v r R V = (Eq 26) where R is the grain radius and r the particle radius. Experimentally, L 3 and (L 3 )α were obtained and used for the grain diameter and particle diameter, respectively; results are shown in Fig. 39. The agreement between calculated and measured grain sizes is considered good. Fig. 39 Comparison of measured and calculated grain size in creep specimens of particulate aluminum- copper alloys. (Ref 23) From the above discussion of grain and particle characteristics, it is evident that there are many points of similarity in their geometrical properties. On the plane of polish, the grain boundaries and particle interphase traces are measured by L A or L p (the perimeter length); the intercept distances for both grains and particles are expressed by L 2 or L 3 ; and the surface area per particle or grain, S/V, and the surface area per unit volume of specimen, S V , apply equally to both volume elements. However, because the grains are space filling, all grain boundaries are shared by two contiguous grain faces; particles, on the other hand, do not usually occupy 100% of the alloy. Therefore, sharing of particle boundaries does not occur as often. To emphasize these differences, Table 3 summarizes the pertinent equations for planar figures, area-filling and separated; the same information for grains and particles is in Table 4. In general, the quantities in the second and third columns of each table are double those in the first column, except for the P L measurements. [...]... 0.0 63 1-0 .0501 1.000 0-0 . 631 0 104 2 0.050 1-0 . 039 8 0. 631 0-0 .39 81 161 3 0. 039 8-0 . 031 6 0 .39 8 1-0 .2512 2 53 4 0. 031 6-0 .0251 0.251 2-0 .1585 230 5 0.025 1-0 .0199 0.158 5-0 .1000 138 6 0.019 9-0 .0158 0.100 0-0 .0 631 69 NA, per mm2 955 Table 8 Calculated distribution of ferrite grain sizes Class interval Diameter of particles, Dj, mm No of grains per mm3, (NV)j 1 0.0 631 27 13 2 0.0501 434 1 3 0. 039 8 831 3 4 0. 031 6 7 630 5... 1.000 0-0 . 631 0 2 0.79 43 0. 631 0-0 .39 81 3 0. 631 0 0 .39 8 1-0 .2512 4 0.5012 0.251 2-0 .1585 5 0 .39 81 0.158 5-0 .1000 6 0 .31 62 0.100 0-0 .0 631 7 0.2512 0.0 63 1-0 . 039 8 8 0.1995 0. 039 8-0 .0251 9 0.1581 0.025 1-0 .0158 10 0.1259 0.015 8-0 .0100 11 0.1000 0.010 0-0 .00 63 12 0.0794 0.006 3- 0 .0040 Because the section area is specified in terms of the largest section area, many sections must be examined to obtain the correct volume. .. zirconium and its alloys, high-speed tool steels, and austenitic stainless steel weldments Examples of etched and heat-tinted specimens are shown in Fig 31 , 32 , 36 , 37 , and 69 in the section "Atlas of Color Micrographs" in this article Examples of attack-polished and heat-tinted specimens as viewed under polarized light and differential interference contrast are shown in Fig 33 , 34 , 35 , and 70 Color... martensite; immerse 2 min 8-1 5 g Na2S2O5, 100 mL H2O Darkens as-quenched martensite; immerse approximately 20 s 3- 1 0 g K2S2O5, 100 mL H2O Darkens as-quenched martensite; immerse 1-1 5 s 1 g Na2MoO4, 100 mL H2O Beraha's tint etch for cast iron and steels; add HNO3 to pH 2. 5-4 .0 (approximately 0.4 mL); immerse 2 0 -3 0 s for cast iron, Fe3P and Fe3C, yellow-orange and ferrite, white; for low-carbon steel add 0.1... 100 mL stock solution plus 0. 6-1 .0 g K2S2O5 (3) Optional additions: 1 -3 g FeCl3, 1 g CuCl2 (cupric chloride), or 2-1 0 g NH4HF2 50 mL saturated Na2S2O3(c), 1 g K2S2O5 Klemm's I tint etch; good for many alloys; immerse 3 min or more for β-brass, - brass, and bronzes; use 1 0-6 0 min for α-brass; use 4 0-1 00 s for coloring ferrite in steels; reveals phosphorus segregation and overheating; longer time produces... microstructures with particles of known shape when other techniques are not feasible Table 9 Properties of a sphere, truncated octahedron, and convex particles Property Sphere, D = 2r Truncated octahedron, edge length = a General equations for convex particles V 4 π r3 /3 11 .31 4a3 V = A'L3 = AH' S 4 π r2 26.785a2 S = 4A' = 4V/L3 A' π r2 6.696a2 A' = S/4 = V/L3 H' 2r 3. 0a H' = V/A = A'L3/A A 2 π r2 /3. .. 30 s for zinc alloys 50 mL Na2S2O3, 5 g K2S2O5 Klemm's II tint etch; immerse 6 min or more for α-brass; immerse 3 0-9 0 s for steels; reveals phosphorus segregation; good for austenitic manganese alloys; immerse 6 0-9 0 s for tin and its alloys 5 mL Na2S2O3, 45 mL H2O, 20 g K2S2O5 Klemm's III tint etch; immerse 3- 5 min for bronze; immerse 6-8 min for Monels 240 g Na2S2O3, 30 g citric acid, 24 g Pb(C2H3O2)2,... i - 1 = 4, i - 2 = 3, i - 3 = 2, and i - 4 = 1 To show how the calculations are made, (NV)4 will be determined from the data given in Table 7 The equation obtained in this case for j = 4 (= i) is: ( N v )4 = 1 [1.65( N A )4 − 0.456( N A )3 − 0.116( N A )2 − 0.0415( N A )1 ] D4 (Eq 29) Substituting the experimental data, ( N v )4 = 1 [1.65( 230 ) − 0.456(2 53) − 0.116(161) − 0.0415(104)] = 7 630 mm 3 0. 031 6... Nv ) j i (Eq 33 ) Therefore, from the data in Table 8, D = 0. 039 3 mm, σ(D) = 0.012 mm, and NV = 27 000 mm -3 An alternative is to plot the cumulative percentages of (NV)j versus particle diameter on log probability graph paper If the size distribution conforms to the log normal distribution, as most particle and grain-size distributions do, a straight line will result Then the values of D and σ(D) can...Table 3 Equations for two-dimensional planar figures Source: Ref 1 Table 4 Equations for three-dimensional grains and particles Source: Ref 1 The parameters defined in Tables 3 and 4 apply equally to interpenetrating two-phase structures and to simple particulate systems In one study a series of beryllium-aluminum alloys (similar to the alloy shown in Fig 38 ) was investigated for . 1 0.0 63 1-0 .0501 1.000 0-0 . 631 0 104 2 0.050 1-0 . 039 8 0. 631 0-0 .39 81 161 3 0. 039 8-0 . 031 6 0 .39 8 1-0 .2512 2 53 4 0. 031 6-0 .0251 0.251 2-0 .1585 230 5 0.025 1-0 .0199 0.158 5-0 .1000 138 6. 1.0000 1.000 0-0 . 631 0 2 0.79 43 0. 631 0-0 .39 81 3 0. 631 0 0 .39 8 1-0 .2512 4 0.5012 0.251 2-0 .1585 5 0 .39 81 0.158 5-0 .1000 6 0 .31 62 0.100 0-0 .0 631 7 0.2512 0.0 63 1-0 . 039 8 8 0.1995 0. 039 8-0 .0251 9 0.1581. No. of grains per mm 3 , (N V ) j 1 0.0 631 27 13 2 0.0501 434 1 3 0. 039 8 831 3 4 0. 031 6 7 630 5 0.0251 33 59 6 0.0199 491 N V , per mm 3 26,847 It may be useful to express the size